Two-station analysis of passive surface waves with continuous wavelet transform and plane-wave-based beamforming

2021 ◽  
pp. 104526
Author(s):  
Yingwei Yan ◽  
Jing Li ◽  
Nan Huai ◽  
Jianbo Guan ◽  
Hui Liu
Keyword(s):  
Wavelet Transform ◽  
Plane Wave ◽  
Surface Waves ◽  
Continuous Wavelet Transform ◽  
Continuous Wavelet

scholarly journals Application of an Automatic Noise or Signal Removal Algorithm Based on Synchrosqueezed Continuous Wavelet Transform of Passive Surface Wave Imaging: A Case Study in Sichuan, China

2021 ◽  
Vol 11 (24) ◽  
pp. 11718
Author(s):  
Jie Fang ◽  
Guofeng Liu ◽  
Yu Liu
Keyword(s):  
Wavelet Transform ◽  
Surface Wave ◽  
Surface Waves ◽  
Continuous Wavelet Transform ◽  
Cross Correlation ◽  
Underground Structure ◽  
Inversion Method ◽  
Continuous Wavelet ◽  
Dispersion Measurement ◽  
Wave Imaging

Passive surface wave imaging based on noise cross-correlation has been a research hotspot in recent years. However, because randomness of noise is difficult to achieve in reality, prominent noise sources will inevitably affect the dispersion measurement. Additionally, in order to recover high-fidelity surface waves, the time series input during cross-correlation calculation is usually very long, which greatly limits the efficiency of passive surface wave imaging. With an automatic noise or signal removal algorithm based on synchrosqueezed continuous wavelet transform (SS-CWT), these problems can be alleviated. We applied this method to 1-h passive datasets acquired in Sichuan province, China; separated the prominent noise events in the raw field data, and enhanced the cross-correlation reconstructed surface waves, effectively improving the accuracy of the dispersion measurement. Then, using the conventional surface wave inversion method, the shear wave velocity profile of the underground structure in this area was obtained.

scholarly journals Continuous Wavelet Transform of Schwartz Distributions in DL2′(Rn), n ≥ 1

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1106
Author(s):  
Jagdish N. Pandey
Keyword(s):  
Wavelet Transform ◽  
Function Space ◽  
Continuous Wavelet Transform ◽  
Inversion Formula ◽  
Continuous Wavelet ◽  
The Real ◽  
Schwartz Distributions ◽  
Distributional Sense ◽  
Wavelet Inversion

We define a testing function space DL2(Rn) consisting of a class of C∞ functions defined on Rn, n≥1 whose every derivtive is L2(Rn) integrable and equip it with a topology generated by a separating collection of seminorms {γk}|k|=0∞ on DL2(Rn), where |k|=0,1,2,… and γk(ϕ)=∥ϕ(k)∥2,ϕ∈DL2(Rn). We then extend the continuous wavelet transform to distributions in DL2′(Rn), n≥1 and derive the corresponding wavelet inversion formula interpreting convergence in the weak distributional sense. The kernel of our wavelet transform is defined by an element ψ(x) of DL2(Rn)∩DL1(Rn), n≥1 which, when integrated along each of the real axes X1,X2,…Xn vanishes, but none of its moments ∫Rnxmψ(x)dx is zero; here xm=x1m1x2m2⋯xnmn, dx=dx1dx2⋯dxn and m=(m1,m2,…mn) and each of m1,m2,…mn is ≥1. The set of such wavelets will be denoted by DM(Rn).

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